‼️ Updated TOP PROBLEMS section ‼️ Added more problems and filled in "coming soon"
AP Physics NotesTable of Contents2/3: Motion & KinematicsOne-Dimensional MotionProjectile MotionRelative Velocity4: The Laws of MotionTypes of ForcesNewton's First Law of MotionNewton's Second Law of MotionNewton's Third Law of MotionFree Body Diagrams & Sums of ForcesFriction5: Work and EnergyWorkEnergySpring/Elastic Potential EnergyPower6: Momentum & CollisionsMomentumImpulseConservation of MomentumTypes of CollisionsGlancing Collisions7: Circular Motion and the Law of GravityAngular DisplacementAngular VelocityAngular AccelerationRelations Between Angular & Linear QuantitiesCentripetal Acceleration/Centripetal ForceUniversal Law of GravitationGravitational Potential EnergyEscape VelocityKepler's Laws8: Rotational Equilibrium & Rotational MechanicsTorqueCenter of Gravity/Center of MassRotational KinematicsRelationship Between Torque and Angular AccelerationMoment of InertiaMoments of Inertia of Extended ObjectsRotational Kinetic EnergyAngular MomentumVector Nature of Angular Quantities9: Solids & FluidsStates of MatterDensityPressureVariation of Pressure with DepthHydraulicsPressure MeasurementsBuoyant Forces & Archimedes' PrincipleFluids in MotionMass/Volume Flow Rate & the Equation of ContinuityBernoulli's Equation10: Thermal PhysicsThe Zeroth Law of ThermodynamicsHeat vs. TemperatureKelvin and Absolute ZeroGas LawsBoltzmann Ideal Gas LawThe Kinetic Theory of GasThe Internal Energy of a Monatomic Ideal Gas11: HeatHeat/Thermal EnergySpecific HeatConservation of Heat Energy: CalorimetryLatent Heat and Phase ChangesTransfer of Heat12: The Laws of ThermodynamicsThe Basics of ThermodynamicsHeat and Internal EnergyWork and HeatThe First Law of ThermodynamicsHeat Engines and The Second Law of ThermodynamicsEfficiencyReversible and Irreversible ProcessesThe Carnot EngineEntropy and Disorder13: Vibrations and WavesHooke's LawGeneral Waves InformationElastic Potential EnergyVelocity as a Function of PositionComparing Simple Harmonic Motion with Uniform Circular MotionPeriod and FrequencyPosition, Velocity, and Acceleration as a Function of TimeMotion of a PendulumDamped OscillationsTypes of WavesFrequency, Amplitude, and WavelengthWave Speed Under TensionSuperposition and InterferenceReflection of Waves15: Electric Forces and Electric FieldsProperties of Electric ChargesInsulators and ConductorsCharging by Conduction (Contact)Charging by InductionCoulomb's LawThe Superposition PrincipleElectric FieldsElectric Field LinesConductors in Electrostatic Equilibrium16: Electrical Energy and CapacitancePotential Difference, Electric Potential, and VoltageElectric Potential EnergyElectric PotentialPotentials and Charged ConductorsThe Electron-VoltEquipotential SurfacesCapacitance and Parallel-Plate CapacitorsCapacitors in ParallelCapacitors in SeriesEnergy Stored in a Charged Capacitor17: Current and ResistanceElectric CurrentCurrent and Drift SpeedResistance and Ohm's LawResistivityTemperature Variation of ResistanceSuperconductorsElectrical Energy and Power18: Direct Current CircuitsEMFResistors in SeriesResistors in ParallelKirchhoff's Rules and Complex DC CircuitsRC CircuitsHousehold CircuitsThe Measurement of Current and VoltageElectricity Review Material19: MagnetismMagnetic FieldsMagnetic Force on a Current-Carrying ConductorTorque on a Current Loop and Electric MotorsMotion of a Charged Particle in a Magnetic FieldMagnetic Field of a Long, Straight Wire and Ampere's LawMagnetic Force Due to Wires With CurrentMagnetic Domains20: Induced Voltages and InductanceInduced EMF and Magnetic FluxFaraday's Law of InductionMotional EMF21: Electromagnetic WavesThe TransformerMaxwell's PredictionsProduction of Electromagnetic Waves by an AntennaProperties of Electromagnetic WavesThe Spectrum of Electromagnetic WavesMagnetism Review Material22: Reflection and Refraction of LightThe Nature of LightReflection of LightRefraction of LightDispersion and PrismsTotal Internal Reflection23: Mirrors and LensesFlat MirrorsImages Formed by Spherical Mirrors (Concave Mirrors)Ray Diagrams for MirrorsAtmospheric RefractionThin LensesCombination of Thin LensesAberrations24: Wave OpticsConditions for InterferenceYoung's Double Slit ExperimentPhase Change due to ReflectionInterference in Thin FilmsNewton's RingsDiffractionSingle-Slit DiffractionSingle Slit vs. Double Slit Interference PatternsDiffraction GratingOptics Review Material26: Special RelativityThe Principle of Galilean RelativityThe Speed of LightEinstein's Principle of RelativityTime DilationSimultaneityLength ContractionRelativistic MomentumRelativistic Addition of VelocitiesRelativistic Energy and the Equivalence of Mass and EnergyEnergy and Relativisic Momentum27: Quantum PhysicsPlanck's HypothesisThe Photoelectric Effect and the Particle Theory of LightX-RaysDiffraction of X-Rays by CrystalsThe Compton EffectThe Dual Nature of Light and MatterThe Davisson-Germer ExperimentThe Uncertainty Principle28: Atomic PhysicsEarly Models of the AtomAtomic SpectraAbsorption SpectraThe Bohr Theory of HydrogenEmission SeriesBohr's Correspondence Principle29: Nuclear PhysicsSome Properties of NucleiCharge and MassNuclear StabilityBinding EnergyRadioactivityThe Three Types of RadiationThe Decay Constant and Half LifeThe Decay ProcessAlpha DecayBeta DecayGamma DecayTop ProblemsChapter 2/3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13Chapter 15Chapter 16Chapter 17Chapter 18Chapter 19Chapter 20Chapter 21Chapter 22Chapter 23Chapter 24Chapter 26Chapter 27Chapter 28References
This course works with only constant acceleration and, therefore, only linear or constant velocity.
Velocity is equal to displacement (change in position) over time.
Acceleration is equal to change in velocity over time.
When an object moves with constant acceleration, the average acceleration equals the instantaneous acceleration. Therefore,
This makes sense, as with linear velocity, the final velocity over a given time period will equal the velocity at the starting point, plus the change in velocity (
With linear velocity, average velocity is equal to displacement over time, but it is also equal to the average of the initial and final velocities.
Going back to (1), we can conclude that
This can be rewritten as:
This intuitively makes sense, as when calculating for the area under underneath a linear velocity graph segment (which is displacement), this equation gives the formula for that area.
If we go back to (3), we can substitute in
Simplified and standardized, this becomes:
This can also be derived using simple integration, which makes more practical sense:
We can also easily visualize it with the following standard gravity function that describes position as a function of time for a projectile:
Notice how this exactly describes (6).
The following equation can be derived to get an equation that does not use time.
This equation feels arbitrary because it is arbitrary. It doesn't make practical sense and it cannot be intuitively visualized. It is purely used to make calculations easier. It can, however, be completely avoided. The following problem will first be solved without the equation, then it will be solved with the equation:
A jet is taking off from the deck of an aircraft carrier. Starting from rest, the jet is catapulted with a constant acceleration of
Without the use of (8) (notice how this process involves solving for time; (8) skips this step):
With the use of (8):
X Component
Y Component
Using these two equations, the path of a projectile can be modeled neglecting air resistance, given the original direction and magnitude of the projectile.
Usually, for projectile motion, the acceleration is gravity.
The velocity of
To figure out the velocity relative to something else, all of the intermediate factors have to be eliminated.
Remember that these are vectors.
Another important thing to keep in mind is how to inverse the relative vectors.
These are the forces that always act upon everything in the universe.
Gravitational Force - attractive force that exists between all objects
Electromagnetic Force - forces that give materials their strength, their ability to bend and squeeze, stretch, or shatter
Weak Force - a form of electromagnetic force involved in the radioactive decay of some nuclei
Strong Nuclear Force - holds the particles in a nucleus together
All types of forces can be classified as one of the two following types.
Contact Forces - forces that result from physical contact between two objects
Field Forces - forces that do not involve physical contact between two objects
Every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.
When the net external force on an object is 0, its acceleration is 0.
Inertia is the tendency of an object to resist any attempt to change its motion.
Mass is a quantitative measurement of inertia
There is not very much difference between an object at rest and one in motion at constant velocity
Force is directly proportional to acceleration.
Mass is inversely proportional to acceleration.
Force is a vector quantity.
Whenever one object exerts a force on a second object, the second object exerts an equal and opposite force on the first object.
Free body diagrams display all forces acting on a single object. Sums of forces equations can then be written based off of free body diagrams to calculate certain values. The axis of free body diagrams can be titled in any way to ensure the most efficient solving process.
Sums of forces equations equal
Let
There are two types of friction:
Static Friction
Represented by
Only applies when the object is not moving
If the external force (applied force) is greater than
Kinetic Friction
Represented by
The object only accelerates if this force is less than the external force; otherwise, it has constant velocity.
Kinetic friction is always less than static friction for the same surfaces and objects.
Both types of frictional forces are proportional to the normal force on the object.
The proportionality constant between friction and normal force is known as the coefficient of friction, and it is denoted by
Energy is the ability to do work.
Energy is never lost, only converted. (This is the law of conservation of energy.)
The unit of work is the joule (J).
If
Kinetic Energy:
Because kinetic energy is the same as the amount of work put into an object's motion if that object was originally at rest (this is due to the law of conservation of energy), we can take the above derived work equation, set
Another intuitive way to derive kinetic energy is using calculus as follows. It is important to note that
Work-Energy Theorem:
Kinetic energy is the work an object is capable of doing in coming to rest.
Potential Energy:
Potential energy is the energy an object has as a result of its position. Because of the law of conservation of energy, we can take the equation
There is an intuitive way of thinking about the potential energy in an object. Imagine raising a ball up to
Potential energy is always relative.
Energy is never lost, only converted:
Energy stored in a compressed spring is called spring or elastic potential energy.
To compress a spring, you must exert a force
Elastic potential energy is defined as the amount of work a spring is capable of doing. Remember that
A revised version of (22) would therefore be:
All this is saying is that all types of energy combined initially is the same as all types of energy combined finally due to the law of conservation of energy.
Including the work done by non-conservative forces which add or take away energy2, we can conclude the following (
A spring-related concept that tends to be confusing is the comparison of the work done by gravity and a spring. For example, take the following problem:
A spring of spring constant
The difficult concept to relate here is: Why does the mass come to a stop when the work done by gravity and the spring are equal?
A common misconception is to assume that the mass would come to rest once the force pulling up and pulling down is the same. With equal forces, however, it is only guaranteed that the acceleration is zero, not the velocity. Therefore, we have to relate something else.
The mass starts at rest and ends at rest, which means that
For a more sophisticated explanation, visit this physics stack exchange page.
Power is the rate at which work is applied. Therefore, the derivative of work with respect to time is power.
Therefore, average power is the average work (or constant work) over change in time.
The units of power is
We can expand (28) to get slightly different forms of the equation:
(30) is the instantaneous power exerted by a force exactly at the time when velocity is
Momentum is inertia in motion.
The units of momentum are
Taking Newton's second law,
And also:
This pretty much means that mass times the change in velocity is the change in momentum. The change in momentum is called impulse.
Impulse:
Impulse can also be defined as a force applied over a certain amount of time. In other words, a force applied to an object for a certain time causes its momentum to change proportionally to the force applied and the time it was applied for.
Given any collision, the momentum before always equals the momentum after. Momentum is always conserved. Kinetic energy, however, is not always conserved. The conservation of kinetic energy is dependent on the types of collisions...
Perfectly Inelastic Collisions
2 objects get stuck together
Momentum is conserved but kinetic energy is not
Example: Dropping a sticky ball that does not bounce at all
(Partially) Inelastic Collisions
2 objects bounce off each other
Momentum is conserved and so is some kinetic energy, but not all kinetic energy
Example: Dropping a ball that bounces back up, but not to the same height you dropped it from
Elastic Collisions
2 objects bounce off each other
Momentum is conserved and so is kinetic energy
Example: Dropping a ball that bounces back up to the same exact height you dropped it from
Most real world collisions fall in the category of partially inelastic collisions.
In elastic collisions, it is important to consider that kinetic energy is completely conserved. This fact allows us to solve problems that would otherwise not be solvable. The following equation can be derived:
All this equation is saying is that the combined velocity of the first object is equal to the combined velocity of the second object.
To envision this, imagine a Newton's Cradle apparatus. Think about the velocities of the 2 spheres on the end before and after the collision. It lines up with (35).
Glancing collisions works with higher dimensions of momentum vectors. This applies when an object hits another one at an angle, and the resulting velocities of both objects are also at different angles.
The general idea is that glancing collisions involve 2-dimensional vectors instead of 1-dimensional vectors.
It is very useful to draw a picture for these types of problems. There are 2 ways to do these types of problems:
Separate the
Represent all momentum quantities as vectors throughout the solving process
Here is an example problem:
At an intersection, a
Solving using strategy 1:
Solving using strategy 2:
Both of the methods work. Which one to use depends on personal preference.
This portion of the course works with radians. Radians have no units, as they are a ratio between arc length and radius.
When rotating a line segment along the origin of a circle, once the distance covered by the end of the line segment is equal to the radius, the angle rotated is one radian.
Using this definition, we can say that:
Remember, radians have no units. Using the above conversion factor, we can convert radians to degrees and vice-versa whenever needed.
Angular displacement is defined as the difference between the original angle and final angle (change in angle) in radians. Angular displacement is
Angular velocity is denoted by
The units of angular velocity are
Angular acceleration is denoted by
The units of angular acceleration are
Arc length is usually denoted by
Going back to the definition of a radian, we can say the angle in radians is equal to arc length over radius because radians are the ratio of arc length and radius. Therefore:
This relates linear displacement and angular displacement. Our next goal is to relate linear velocity to angular velocity. To do this we need to derive
We can use (39) to arrive here as follows:
The next goal is to relate linear acceleration to angular acceleration. To do this we need to derive
All of these derivation have a direct correlation with each other considering the radius.
Here are all the equations listed:
Contrary to popular belief, centripetal force doesn't push outward, it pushes inward.4 The only reason people think it pushes outward is actually because of inertia. The tendency for an object to not move wants to make it leave angular motion, in order for it to stay in angular motion, there must be a force pushing back or an acceleration toward the center of the circle. This is called centripetal acceleration.
A really important concept to understand is that centripetal acceleration/centripetal force is not some external force that just pulls on an object. It is the resulting force of other forces. For example, when a car is traveling in a circle, the friction in the turned wheels causes there to be a force that pushes inward. This is called centripetal force. There isn't just some external force acting on the car making it go in a circle. If the wheels were straight, the frictional force to provide the centripetal force wouldn't be there, and the car would go straight. Similarly, on a banked curve, the
The derivation of centripetal acceleration is a complex one. However, I will try my best to explain. centripetal acceleration is denoted by
Look at the diagram below for reference on the derivation.

Given a change in time,
The magnitude of
As seen on the diagram,
The acceleration we're interested in is
It is really important to note that the acceleration we're interested in is the instantaneous acceleration, therefore
The triangle created by
As
We now have
Since
In our case,
We now have
Here's the purely mathematical way to derive it:
This YouTube video explains how centripetal force works really well.
Centripetal force is is just centripetal acceleration times mass:
Every particle in the universe is attracted to every other particle in the universe. The force of attraction between them is called gravity. The force of gravity is directly proportional to the product of the mass of two object and inversely proportional to the square of the distance between their center of masses:
It might be useful to ask where this equation comes from. Unfortunately, though, this equation doesn't have a derivation.5
With objects near Earth's surface, the equation
The previous equation only described the change in gravitational potential energy. This new definition will get us the total value of gravitational potential energy, which can be defined as: the work needed to bring an object from infinitely far away (where gravity is

Because gravity naturally brings objects close together, the work required by an external force to bring objects together is negative because it happens naturally.
The force of gravity gets exponentially less as distance increases. The work done by gravity is force multiplied by distance. Remember that
The bounds of the integral are from
the work gravity is capable of doing in bringing an object from infinitely far away (Where gravity is 0 and GPE is 0) to a distance
away.
Solving the integral in (47), we get:
At
Therefore, the formula for total GPE is:
This YouTube video explains the derivation really well.
Escape velocity is the lowest possible velocity an object can have in order to escape the gravitational pull of another object.
There are a few things we must note to be able to derive the formula for escape velocity. The total energy of an object is the kinetic energy plus the gravitational potential energy.
As an object moves away, gravity will lessen its velocity, so the kinetic energy will eventually turn into gravitational potential energy. The escape velocity describes the velocity required to completely escape the gravitational pull of an object. In theory, this can only be achieved at an infinite distance.
Therefore, for theoretical purposes, we can assume that the object is capable to moving infinitely far away. At this distance, gravitational potential energy is
Solving for
Notice how escape velocity doesn't depend on the escaping object's mass, only the mass of the object it is escaping from.
This YouTube video explains the derivation.
All planets move in elliptical orbits with the Sun at one of the focal points.
A line drawn from the Sun to any planet sweeps out equal area in equal time intervals.
All this means is that a planet moves faster when it is closer to the Sun but slower when it's farther away.
The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the Sun.
All this means is that the orbital period for every planet is dependent on how far away (on average) it is from the Sun.
This can be derived in equation form.
Going back to centripetal force and acceleration, the sums of forces equation for an orbiting planet would be (
Rearranging, we get:
The proportionality constant can be denoted by
Torque is the ability of a force to rotate a body about some axis. It is measured as the product of the force being applied perpendicularly to the axis of rotation. Torque is denoted by
The units of torque are
The positive/negative signs of torque are consistent with the signs of radians. Moving up from the positive
Sums of torque diagrams and equations can be created in the same manner as sums of forces diagrams and equations to solve for unknowns. In rotational equilibrium, the sums of torque is equal to
An intuitive way to think about torque is imagining opening a door. Pushing closer to the hinges requires more force to achieve the same rotation as pushing farther away from the hinges. Therefore, if the force was the same, you would exert less torque when pushing closer to the hinges because the door wouldn't open as much. If you applied the same force to the edge of the door, the door would move a lot more, so the torque exerted would be greater.
Another way to think of torque is the strength/force of rotation.
When referring to objects with a weight, we usually consider the weight to be concentrated at one point in space. This is called the center of gravity or center of mass.
To calculate where the center of mass is, we have to compute a sort-of average on all the points of mass in a system. To start, pick a point that every other point will be relative to.
The center of mass relative to that point in the
Similarly, the center of mass relative to that point in the
Because of the similarities between linear and rotational quantities, linear kinematic equations hold true for rotational kinematics.
When the net torque on an object is not
Torque is equal to
The moment of inertia is generally thought of as the inertia of rotating objects. For example, a rotating circular object that has more mass located farther away from its origin is harder to start rotating and stop rotating than an object whose mass is closer to the origin. This is measured as the moment of inertia.
Notice the similarities between the two equations:
In the linear equation, the mass is what provides the inertia, but in the angular equation, the moment of inertia is provided by
In order to get an equation to accurately model real world objects where there is mass spread out, we have to sum up every point's moment of inertia. This is denoted by
Even though
The full relationship is:
To calculate moments of inertia for extended objects, you need integral calculus. For this reason, all the formulas of moments of inertia of extended objects will be given on tests etc.
The following part of this subsection is NOT needed for this course.
If you are curious, though, as to how to actually calculate these moments of inertia, I will walk you through it.
Given a finite number of points with finite masses that make up an object we know that the moment of inertia is equal to the sum of them:
Now, if we take an extended object, each point will be infinitesimally small and would, therefore, have an infinitesimally small mass. The radius, however, will stay the same.
We cannot directly relate
For one example, let's calculate the moment of inertia of a rod of length
Because the rod rotates about the center and its length is
To relate mass to linear quantities, we can use the linear mass density of the object, denoted by
Differentiating both sides and keeping in mind that the linear mass density is a constant, we get:
Evaluating the integral, we obtain:
Remember that
This was the derivation of the moment of inertia of a rod of length
Other objects' moment of inertia can be derived in very similar ways. This was just one example.
Remember the original kinetic energy formula,
Converting tangential velocity to angular velocity using
The sum of angular velocities is just
Remember the equation for moment of inertia,
Comparing this to the original kinetic energy equation, this should not be surprising considering the similarities we've seen in the past. Going from translational to rotational,
Keep in mind the similarities between
Because translational momentum is
Unlike translational moment, whose units are
Angular momentum also means there is rotational impulse. It works very similarly to translational impulse.
Replacing
Putting everything together as seen in (78), we obtain:
This is really similar to translational impulse. All this is saying is that, given an applied torque over a period of time, the angular momentum will change accordingly.
Conservation of momentum laws apply. (YouTube video example)
For single point particles with a known tangential velocity and radius, the angular momentum can be found using the following formula:
This is because
Until now, we've considered only the magnitudes of angular quantities like
The direction of this vector can be found using the right hand rule. For this, you must use your right hand. Stick your thumb out and curl the rest of your finger. The direction that the rest of your fingers point should match the direction the object is rotating. The direction your thumb points is the direction of the vector. This works for angular velocity, angular acceleration, and angular momentum
Solids
Definite shape and definite volume
Can be crystalline (atoms with an ordered structure) or amorphous (random atomic structure)
Generally considered not compressible
Liquids
Definite volume but indefinite shape
Intermolecular forces are not strong enough to keep the molecules in fixed positions
Generally considered not compressible
Gases
Indefinite shape and indefinite volume
Gases are compressible
Gases fill up the container that they are in completely
Density is defined as mass per volume. The more mass something has given the same volume, the more dense it is. Density is denoted by
The units of density are
The density of water is known to be
The specific gravity of a substance is the ratio of its density to the density of water at
Pressure is the force that's exerted over an area. The more concentrated the force (so the smaller the area it's spread out over), the greater the pressure. Pressure is denoted by
The units of pressure are
Within a fluid, all points at the same depth must have the same pressure as long as the fluid is in static equilibrium (it's not moving).
Look at the following figure:

The force pushing on top is
Creating a sums of forces equation for the cutout portion of fluid using the free body diagram, we get:
It equals
Going back again to the density equation, we know that
Dividing each term by
Look back at the diagram. This equation states that the pressure between the top and the bottom of a fluid, separated by a height
The shape of the fluid container does not matter for this principle.
Pascal's Principle states that pressure applied to an enclosed fluid transmitted undiminished to every point of the fluid and to the walls of the containing vessel. All this states is that pressure always pushes evenly outward in all directions.
In hydraulics, a U-shaped tube (called a u-tube) is used to be able to apply larger forces when a smaller force is initially put in.
Remember, the pressure at any same depth of a liquid is always the same, no matter the shape of the container.
Look at the following diagram.

Our goal is the measure the pressure of the supplied gas.
Because the pressure at any same depth of a liquid is always the same, we know that the pressure at point
Using the variation of pressure with depth equation, we know that the pressure at the bottom of the displaced liquid is
Therefore:
The force that makes objects float is called a buoyant force. This force exists because pressure is larger at greater depths.
Given an object in a fluid, the pressure pushing downward on top will be less than the pressure pushing upward on the bottom. Therefore, the upward force exceeds the downward force and the object floats.
Because
This means that the buoyant force is equal to the weight of the fluid that is displaced (the space that the object's volume takes up where there would be fluid).
Archimedes' Principle states that:
Below, only the highlighted portions are what we work with in this course. The rest are useful to know for vocabulary.
Types of Fluid Flow
Steady Flow
The velocity of every particle is constant
Unsteady Flow
The velocity of every particle varies over time
Turbulent Flow
An extreme unsteady flow that usually occurs when there are obstacles in the path of a fast-flowing fluid
Compressibility of Fluids
Incompressible Fluids
Most fluids are nearly incompressible - their density remains constant as pressure changes
Compressible Fluids
Gases are compressible
Viscosity of Fluids
Viscous Fluids
Fluids that resist flow, such as honey or oil
Non-viscous Fluids
Readily flowing fluids, such as water
Rotatability of Fluids
Rotational Fluids
When part of a fluid has rotational and translational motion
Irrotational Fluids
When the fluid has only translational motion
An incompressible, non-viscous fluid is called an ideal fluid.
When fluids flow, streamlines are often used to represent the motion of the fluid.8
Steady flow is often called streamline flow.
The equation of continuity states that the mass of fluid entering one end of a vessel must equal the mass exiting the other end of the vessel.9
Mass Flow Rate must remain constant.
Imagine a pipe with fluid flowing through from point
At point
At point
Given a change in time,
At point
At point
Knowing the cross sectional areas:
At point
At point
Because
Knowing the density and volume at point
Knowing the density and volume at point
Dividing both sides of the equations by time, we get
Because the mass flow rate must remain constant, we arrive at:
However, in this course we only work with incompressible fluids, meaning the density always remains constant.
Therefore, (92) can be simplified into
All this states is that given an incompressible fluid, the volume of fluid entering the vessel in a certain time must be the same as the volume of fluid leaving the vessel.
Given an incompressible, non-viscous fluid, whenever a fluid flowing in a horizontal pipe encounters a region of reduced cross sectional area (thinner pipe), the pressure of the fluid drops.
Bernoulli's Principle: Where the velocity of a fluid is high, the pressure is low, and where the velocity of a fluid is low, the pressure is high.10
If the pipe has an elevation difference from low to high, given the width of the pipe does not change, the the pressure at the lower elevation must be greater than the pressure at the higher elevation to be able to keep the fluid moving upward.
Look at the following diagram:

This diagram shows a pipe that has an elevation change and a width change.
Remember that
Also remember that
Now, going back to the figure, notice how, because of the equation of continuity,
This can be considered the non-conservative work done on the fluid.
Remember that
Because
Dividing each side by
Putting all the
This YouTube video dives more in depth into how this equation works and applications of the equation.
Some materials are better than transmitting heat than others. Get a metal tray and a cardboard box out of a freezer. The metals feel colder even though they are at the same temperature. This is because the metal is better at transmitting heat and sucks the heat from your hand faster than the cardboard does in order to achieve thermal equilibrium.11
The Zeroth Law of Thermodynamics states that if bodies
Temperature is a measure of the average kinetic energy of the particles in an object. Heat is the total kinetic energy of the particles in an object.
Heat is a quantity that measures the total energy that is present in an object in the form of particles moving (it is dependent on how much of an object there is) while temperature is local (It doesn't depend on how much of the object you have).
For example, let's say you have a teacup that's filled with boiling water and a bucket that's filled with lukewarm water. The temperature of the teacup is greater because the particles move faster, so their average kinetic energy is greater. However, the heat of the bucket could be greater because, even though the particles move slower, there are a lot more of them, so the total kinetic energy is more.
Normal everyday thermometers are limited by the temperatures that they are capable of measuring. Most of them consist of a tube with a liquid that expands or contracts to display the corresponding temperature. A better type of thermometer uses gas and pressure to determine temperature. These are called constant volume gas thermometers.
The lower the temperature of something becomes, the slower the particles in the object move. At one point, however, they stop moving. At that point, they cannot get any colder. This is a temperature known as absolute zero. Absolute zero is
In order to better be able to make calculations, in science, we often use the Kelvin scale. One Kelvin (
In the imperial unit system, Fahrenheight is used.
Gases that behave ideally make calculation a lot easier. Ideal conditions are low pressure and high temperature. Luckily, this is true for most places on Earth, so the equations hold up well.
Avogadro's Number - The number of items in a mole, denoted by
Boyle's Law - Pressure is inversely proportional to volume.
Boyle made the observation that as you decrease the volume of a gas, the pressure increases:
Charles's Law - Temperature is directly proportional to volume.
Charles made the observation that, as the temperature of a gas increases, so does the volume.
Avogadro's Law - Moles are directly proportional to volume.
Avogadro made the observation that, the more moles you have of something, the more space it takes up.
Combined Gas Law or Ideal Gas Law - a combination of all three laws, which includes moles.
The three proportionalities are:
Combining the three proportionalities into one, we get:
To make them equal, we need a proportionality constant. We will denote this using
Putting this into standard form, we get that the Ideal Gas Law is:
or
Remember that, according to Avogadro's number, the number of moles if equal to the number of items divided by Avogadro's number.
Substituting this in for
Both
Because gases contains millions of molecules, it helps us to look at their average speeds.
For the kinetic theory of gas to work, we make the following assumptions:
The number of molecules is large, and the average separation between them is large compared with their dimensions. This means that the molecules occupy a negligible volume in the container.
The molecules obey Newtons Laws of Motion, but as a whole, they move randomly.
The molecules undergo elastic collisions with each other and with the walls of the container. Thus, in the collisions both kinetic energy and momentum are conserved.
The forces between molecules are negligible except during a collision.
The gas under consideration is a pure substance; that is, all molecules are identical.
Pressure of a gas is caused by its molecules hitting the sides of the container. Given a cubical container whose side length is
While approaching the wall of the container, the particle has a a velocity
Let
Also, remember that
Solving for
Because this is the force exerted by the gas, we know that the container exerts an equal but opposite force. Therefore, the force exerted by the container on the gas is:
Remember that there are
Note that
Remember that
or
Also, remember that
Isolating
The internal energy
This can be rewritten as:
Usually,
Substituting these into the previous equation and simplifying, we obtain:
There are two types of heat, mechanical energy (
Heat/thermal energy is energy that is transferred between a system and its environment because of a temperature difference between them.
One of the most widely used units for heat is
Calories in food are measure by Calories (specifically a capital C).
In this course, however, because heat is a form of energy, we will use
Joule's Experiment
Joule set up an apparatus that would convert mechanical energy into heat by spinning a paddle in water. He observed that the loss of mechanical energy was proportional to the increase in temperature of the water.
Specific heat is the heat required to raise
Given a substance with a specific heat of
When
When
The specific heat of water is
A calorimeter is an insulated container so that heat cannot go in or out. However, heat can flow between materials and objects within the calorimeter.
If not in equilibrium, cold objects heat up and hot objects cool down until thermal equilibrium is reached.
The heat gained equals the heat lost.
A substance usually undergoes a temperature change when heat is transferred, but not always. When the substance is changing from one state of matter to another, it either absorbs or emits heat, but the temperature doesn't change while the phase change is happening.
Here are the types of phase changes:

The following diagram shows the temperature of a substance as a function of heat added:

Notice how, in the flat segments, heat is being added or removed, but the temperature is not changing. The substance is going through a phase change.
When temperature is changing,
Also, note that the energy in the vaporization/condensation phase change is much greater than the energy in the melting/freezing phase change.15
There are 3 ways in which heat can be transferred.
Conduction The process in which heat is transferred directly through materials in contact. The molecules/atoms in the material vibrate and pass some of their energy onto other particles. Thermal conductors conduct heat well (usually metals) while thermal insulators conduct heat poorly.
Convection The process in which heat is carried from place to place by the bulk movement of a fluid. When part of a fluid is warmed, such as the air above a fire, the volume of the fluid expands and moves upward, and colder surrounding fluid replaces it. This flow is called a convection current. Natural convection results from differences in density, such as the air around a fire. Forced convection results from a forced external movement such as one caused by a fan or pump.
Radiation
The process in which heat energy is transferred through the means of electromagnetic waves.
All objects continuously radiate energy. Generally, an object does not emit much visible light until the temperature of the object exceeds
Thermodynamics is the branch of physics that is built upon the fundamental laws of nature that heat and work obey.
In thermodynamics, the collections of objects upon which attention is being focused is called the system while everything else is called the surroundings.
Walls that allow heat to flow are called diathermal and ones that do not are called adiabatic.
Internal energy refers to all of the energy belonging to a system while it is stationary (no kinetic energy), including heat, nuclear, chemical, and strain (stretch or compression) energy.
Thermal energy is the portion of internal energy that changes when the temperature of a system changes.
Heat transfer is a process caused by a temperature difference between a system and its surroundings.
Consider an ideal gas contained in a cylinder with a movable piston on top, as the following diagram shows:

In equilibrium, the gas has a volume of
If the piston has a cross-sectional area of
If the piston is pushed down a distance of
All this is saying is that the work done on the gas is equal to the product of the pressure of the gas and the volume that it is changed.
This equation strictly represents the work done ON the gas and is highly dependent on the direction it is compressed.
If the gas expands, then
If the gas is compressed, then
The above equation only works if the pressure of the gas remains constant. When it does, it is called an isobaric process.
In the past, we've only considered mechanical energy. However, we will now include internal energy.
Energy can be transferred between a system and its surroundings in two ways:
Via work done by or on the system
Via heat transfer
Suppose that a system gains heat and does no work. The internal energy, denoted by
Similarly, suppose that a system does
The first law of thermodynamics states that, if a system undergoes a change from an initial state to a final state, where
The second law of thermodynamics states that heat flows spontaneously from higher temperature to lower temperature.
A heat engine is any device that uses heat to perform work, as shown in the following diagram:

Here's how it works:
In the boiler, liquid water in converted into steam
The steam passes through the intake valve into the cylinder and expands against the piston, forcing it to move
The work done on the piston comes from the energy of the steam, so, by moving the piston, the steam loses energy, cooling off, and letting the piston drop back
The steam is forced through the exhaust valve, into the condenser, where it is reconverted into water
An important characteristic of an engine is its efficiency. An engine that converts most of the input heat into work is efficient. Efficiency can be measured using the ratio between work produced and inputted heat. Efficiency is denoted by
Efficiency is often measured in percent.
In an engine, the input heat,
The work the engine provides can be expressed by difference in heat between the hot and cold reservoirs.
This can also be substituted into the efficiency equation to get:
A reversible process is one that can be performed so that, at its conclusion, both the system and surroundings have been returned exactly to their initial conditions, including energy. An irreversible process is one that does not satisfy these requirements.16
Sadi Carnot proposed that a heat engine has a maximum efficiency when the processes within the engine are reversible. This is known as Carnot's Principle.
Carnot's Principle: No irreversible engine operating between two reservoirs at constant temperatures can have a greater efficient than a reversible engine operating between then same temperatures. Furthermore, all reversible engines operating between the same temperatures have the same efficiency.
To describe the Carnot cycle, we assume that the substance working between the input temperature,

This diagram illustrates the four phases of the Carnot cycle.
Process
Process
Process
Process
For information about PV diagrams and more on the work-heat-energy relationship of gasses, watch NOTES SMAY CH 12 Day 2 Finish 1st Law and Carnot Conceptual VIDEO starting from 23:59.
The difference in
Carnot showed that the thermal efficiency of a Carnot engine is the following, where
To maximize efficiency,
In the 1800s, Rudolf Clausius created a concept called entropy, which is a measure of how disordered the universe is. "Disorder" is a measure of the usable energy in the universe and how spread out all of the energy is. A disordered arrangement is much more probable than an ordered one. Isolated systems tend toward greater disorder where everything is spread out.
Think about a block of ice that you took out of the freezer. For all the cold to be concentrated in the ice at any given moment in time is a very unlikely scenario, and, therefore, the heat starts spreading out until the ice melts and everything is the same. Unlikely scenarios tend toward likely and predictable ones.
You can, however, create processes that seem like they decrease entropy. For example, using the AC. The AC makes a cool house cooler, making the system more ordered. The reason this is possible is that, even though that specific action causes entropy to decrease, the processes that make the AC work (power lines, generators, power plants, etc.) increase entropy by more than the AC decreased it. Overall, there is still an increase in entropy.
An interesting way to think about entropy is using the Sun as an example. What do we get from the Sun? The obvious answer to this question is energy. However, the Earth also expels the same amount of energy back into space that it gets from the Sun. If it didn't, then the atmosphere would heat up and life wouldn't be possible. So knowing that the net energy change the sun provides is zero, what do we actually get from the sun?
The answer to this question is that we get low-entropy energy or ordered energy – usable energy that we then convert to high entropy energy that is no longer useful.
Every process that ever happens in the universe causes the total entropy of the universe to increase. This means that entropy is always increasing. The only process that doesn't cause entropy to increase are reversible processes, which are not really possible.
As current theories predict, in the extremely far future, after an unimaginable amount of time has passed, everything in universe will be so spread out that nothing interesting will happen again, and entropy will reach a maximum. This is known as the heat death of the universe.
Entropy is denoted by
Consider a reversible process.
It can also be written as:
When an irreversible process occurs and the entropy of the universe increases, the energy available for doing work decreases.
There are four types of waves (periodic motion):
Sound waves
Water waves
Seismic waves
Electromagnetic waves (light waves)
Seismic waves are the only ones we will not be working with in this course.
Remember the equation that models the force of a spring:
Simple Harmonic Motion and Hooke's Law - SHM occurs when the net force along the direction of motion obeys Hooke's Law: the net force is proportional to the displacement from the equilibrium point and is always directed toward the equilibrium point.
Amplitude (
Period (
Frequency (
Furthermore, the acceleration of the target object in SHM can be solved for using a combination of Hooke's Law (
Remember the equation for elastic potential energy. Elastic potential energy is also defined as the amount of work a spring is capable of doing. Remember that
Solving the integral, we get:
Considering a spring with an object stretched to length
Solving the equation for
This equation is only useful because it will be used in a future derivation. Just know conservation of energy, and specifically:
During the motion of the spring and object, the energy will be a mix of elastic and kinetic energy.
Picture a turntable with a ball on the edge. As the turntable rotates, if we shine a light downward, the ball moves with constant velocity, and the shadow oscillates back and forth with simple harmonic motion. How do we know for sure, though?
Remember that
Look at the following diagram:

Look at the two similar triangles, we know that
and
Because these are equivalent expressions, we can set them equal to each other and obtain:
The
Period (
We can use these values to express the velocity:
And also the time/period:
Consider a quarter cycle, where the energy in the cosine wave turns from spring potential energy into kinetic energy:
Substituting this into the period equation, we get:
Notice that the period does not depend on
The angular frequency,
This type of movement is known as rectilinear motion. Picture a
The default equation has an amplitude of
The amplitude can be modified using a leading coefficient:
Because this is a function of time, we can find the first and second derivatives with respect to
Notice how these equations relate to period and angular frequency.
Think about a spring that has a mass
isolating
Remember that
Notice the similarities between this and:
Looking at this form, we know that the following relationship it true:
We can now obtain our angular frequency formula:
And because we know that
Pendulums oscillate back and forth, too, similar to SHM, but is it actually SHM?
Consider a pendulum with a length
This equation is not in the form
However, for angles smaller than about
Going back to out period equation, substituting
The describes the period of a pendulum.
Based off of our equations, turntables would turn forever, and pendulums would swing indefinitely. However, in reality, friction and air resistance causes these contraptions to stop oscillating over time.
Waves are always caused by some sort of vibrating object. The mechanical waves discussed in this chapter require:
A disturbance
A medium that can be disturbed
A physical connection or mechanism through which adjacent portions of the medium can influence each other
Transverse Wave - A traveling wave where disturbance is perpendicular to the direction of travel
Longitudinal Wave - A wave where disturbance is parallel to the direction of travel
Wave speed depends on the tension in the string. If a string under tension is pulled sideways and released, the tension,
The wave speed is inversely dependent on the mass per unit length,
The exact relationship is:
This means that the wave speed depends on only the properties of the string, not the amplitude.
If you're curious as to how exactly you derive this formula, check out this YouTube video.
Two traveling waves can meet and pass through each other without being altered or destroyed.
The superposition principle states that when two or more traveling waves encounter each other while moving through a medium, the resultant wave if found by adding together the displacements of the individual waves point by point.
Constructive Interference When two waves add up to form a bigger resulting wave
Destructive Interference When two waves add up to form a smaller resulting wave or even cancel out completely
When a wave encounters a boundary, like a wall, it is reflected. There are two possible scenarios that can be thought of using a rope:
The end of the rope is fixed to the wall If the end is fixed, the wave will not only bounce back, but it will invert and flip over.
The end of the rope is free to move (eg. the end is a loop tied around a pole) If the end is free to move, the wave will still bounce back, but it will not invert or flip.
Around 700 B.C., the Ancient Greeks conducted the earliest known study of electricity. They noticed that small objects would be attracted to amber if it was rubbed with wool.
These electric charges are caused by a difference of electrons. In a conductive object, such as metal. Electrons are allowed to pass freely among the particles that make up the metal. This is known as a "sea of electrons".
Electrons are negatively charged, and so there are two possible states that a material can have:
Positive Because electrons are negatively charged, if an object loses electrons, it becomes positive charged.
Negative For the same reason, if an object gains electrons, it becomes negatively charged.
The most important law about these electrons is that like charges repel and unlike charges attract. Specifically, here is a table of all possible combinations (Ø means neutral):
| Charge 1 | Charge 2 | Attraction |
|---|---|---|
| + | - | Attract |
| + | + | Repel |
| - | - | Repel |
| + | Ø | Attract |
| - | Ø | Attract |
| Ø | Ø | No attraction |
Two objects attract when there is a difference in type of charge. Otherwise, they repel.
Electric forces are stronger than gravitational forces. If you rub a balloon on the carpet, the rubber will become electrically charged and it will stick to the wall.
Most substances are electrically neutral.
Particles become electrically charged by gaining or losing electrons.
Electric charge is always conserved.
In 1909, Robert Milikan proposed that if an object is charged, its charge is always a multiple of a fundamental unit of charge; it is quantized. This fundamental unit of charge is equivalent to the charge that one electron carries, denoted by
In conductors, electric charges move freely in response to an electric force. All other materials are called insulators.
Conductors They let electrons flow freely ("sea of electrons").
Insulators They hold onto electrons tightly.
Semiconductors They have properties that are somewhere in between conductors and insulators, such as Si or Ge on the periodic table (we do not use semiconductors in this course).
When a neutrally charged object comes into contact with a charged one, the amount of charge "equalizes" between the two objects. Meaning that if a neutral object comes into contact with a negatively charged object, it also becomes negatively charged. The object doing the charging remains negatively charged, but to a lesser degree. (Works the same for positive)

Look at the diagram above. The Metal sphere is originally neutrally charged. When the negatively charged ebonite rod is held near it, the electrons escape to as far away from the rod as possible because they repel, making the side closer to the rod positively charged. When a grounding wire is introduced, the electrons have an escape route and move even farther away from the negatively charged rod, leaving the sphere. If the grounding wire is removed, then the sphere remains positively charged.
In 1785, Charles Coulomb established the fundamental law of electric force between two stationary charged particles.
This equation is analogous to the Universal Law of Gravitation.
The magnitude of the electric force,
This force is attractive if the charges have opposite signs and repulsive if the charges have the same sign.
Also, the electric forces between elementary particles are far stronger than the gravitational forces between them.
When a number of separate charges act on the charge of interest, each exerts an electric force. These electric forces can all be computed separately, one at a time, then added as vectors.
Remember that field forces are types of forces that don't need contact between objects: gravity and electricity. A person named Michael Faraday proposed the idea of an electric field. It is a field, analogous to a gravitational field that is said to exist around a charged object.
The definition of an electric field is the amount of force per coulomb of charge that would be exhibited on a positively charged particle at any given point.
Think back to gravity. A planet, for example, exhibits a gravitational field around it. Any object within the gravitational field is subject to a force per kilogram of mass just like for an electric field. Gravitational fields deal with the mass of objects, while electric fields deal with the charge of objects.
An electric field is analogous to the acceleration caused by gravity.
An electric field produced by a charge
Current topic equations will be on the left, while analogous gravity equations will be on the right.
This means that:
Now, look at the two force law equations:
Substituting, we get:
The final electric field equation is:
All of these equations and ideas are exactly analogous to gravity.
Relationship Between Electric Fields and Distance:
This is an alternate equation for electric fields.
A convenient way to visualize electric field patterns is to draw lines pointing in the direction of the electric field vector at any point.
The electric field vector
The number of lines per unit area through a surface perpendicular to the lines is proportional to the strength of the electric field at a given region.
The magnitude of
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Rules For Drawing Electric Field Lines
Lines begin on positively charged particles and end on negatively charged particles unless they approach infinity.
The number of lines drawn leaving or ending a particle is proportional to the strength of the charge.
Field lines cannot cross.
Examples of Electric Field Lines

When no net motion of charge occurs within a conductor, the conductor is said to be in electrostatic equilibrium.
The electric field is zero everywhere inside the conducting material.
Any excess charge on an isolated conductor resides entire on its surface, as free-flowing electrons attempt to get as far away from each other as possible.
The electric field just outside a charged conductor is perpendicular to the conductor's surface.
On an irregularly shaped conductor, the charge accumulates at sharp points, where the radius of curvature of the surface is the smallest.
The electric potential inside a conductor is constant and equal to the electric potential at its surface.
Electric potential or Volts is a property of space that describes how much electric potential energy could be present at a given point per unit of charge.
Voltage is electric potential difference, represented by
For example: if point
The work done by an electric field is
As the electric field does work,
Putting everything together we obtain:
It is also important to note, units-wise, that
In electric circuits, points at which the potential is zero is often defined by grounding some point in the circuit. For a point charge, however, similar to gravity, a point of zero potential can be defined as being an infinite distance away from the charge. The same goes for electric potential energy. Electric potential energy can be derived as follows:
Electric potential energy can be defined as the work needed to move a positive particle toward another positive particle with an electric field

Because the direction of force needed is to the right, and the electric field pushes the other way, the the force by the electric field is
Because
Remember the formula:
Solving this integral, we get
Remember that the electric potential is
Going back to the electric potential energy formula and plugging it in, we get:
Simplifying and converting to standardized variables, we get the formula for electric potential:
No work is required to move a charge between two points that are the same potential.
The electric potential is a constant everywhere on the inside of a charged conductor and equal to the value at its surface.
The potential inside a conductor is not necessarily zero, even though the interior electric field is zero.
One electron-volt,
An equipotential surface is one on which all points are at the same potential. No work is required to move a charge at a constant speed on an equipotential surface. Equipotential lines (isolines) are always perpendicular to electric field lines.
A capacitor can be charged in order to store energy in circuits that can later be reclaimed for use. A diagram of a typical capacitor is shown below.

The insulator in the middle (usually air) gets charged in a way where electricity can be released from the capacitor at a later time. Current does not pass through a capacitor. The charge on the one plate induces the charge on the other plate.
The capacitance,
The SI unit of capacitance is the farad (
The capacitor diagram shown above is what is known as the parallel plate capacitor. The electric diagram version of the one shown above is (the short side of the battery is the negative side):

For a parallel plate capacitor:
The capacitance is directly proportional to the area of one of the plates (both plates have equal area)
The capacitance is inversely proportional to the distance between the plates.
Two constants are also involved:
Putting everything from above together, we get the following equation for capacitance:
It is important to note that the electric field in between the plates is nearly constant.
Two capacitors in parallel in a circuit are arranged as follows.

Capacitors in parallel both have the same voltage across them. The charges of the capacitors, however, add up:
For Capacitors in Parallel:
Voltage is the same
Charge adds up
For
The equivalent capacitance of a parallel combination of capacitors in greater than any of the individual capacitances. This is true because capacitors in parallel is like increasing the area of a single capacitor.
Two capacitors in series are arranged as follows:

Capacitors in series both have the same charge across them. The voltages of the capacitors, however, add up:
For Capacitors in Series:
Charge is the same
Voltages add up
For
The equivalent capacitance of a series combination is always less than any individual capacitance in the combination. This is true because capacitors in series are like increasing the distance between plates.
Capacitors can store electrical energy, and that energy is the same as the work required to move charge onto the plates. As soon as the capacitor's plates touch, or have have a conductor placed between them, the electricity will flow.
Because the units for
Solving this integral, we get that the energy,
Current is the rate at which charge flows. Suppose
The SI unit of current is the Ampere (
Conventional current is defined by the direction that positive charges flow.
As charges move along a wire, electric potential is continually decreasing.
In a wire with a cross-sectional area of
Charge carries move with a constant average speed called the drift speed.
Substituting this into the previous equation, and dividing by
When a voltage is applied across the ends of a metallic conductor, we find that the current is proportional to the voltage.
This can be rewritten as
Resistance is defined as the ratio of voltage across the conductor to the current it carries:
The SI unit for resistance is Ohms (
The resistance of a conductor increases with length, and decreases with area. Using a proportionality constant
This proportionality constant is called the resistivity of the material.
Good conductors have low resistivity
Good insulators have high resistivity
For most metals, resistivity increases with increasing temperature. As the heat goes up, molecular motion also goes up and it becomes harder for an electron to smoothly make its way through the material. The following equations describe the linear relationship between resistivity/resistance and temperature (
Superconductors are a class of metals and compounds with resistances that fall virtually to zero below a temperature called the critical temperature. This temperature is usually extremely low.
The chemical energy that's stored in a battery is continuously transformed into thermal energy (through collisions). These collisions are what cause a lightbulb to heat up and glow.
As a charge
The power, representing the rate at which energy is delivered to the resistor, is therefore:
This equation can be slightly modified using
EMF stands for electromotive force. It represents a voltage. In a battery, the EMF is the "theoretical" voltage. In reality, by the time the current comes out, it experiences some internal resistance in the battery. This is the "actual" voltage of the battery.
For example, a battery may be labeled as
EMF is represented by
With a current
Consider an external resistor with resistance
Soling for current:
We generally assume the internal resistance of a battery is zero unless otherwise specified.
Resistors in series and in parallel work very similar to capacitors in series and parallel, so I will not include the repetitive derivations.
Two resistors in series in a circuit are arraganged as follows:

For Resistors in Series:
Current is the same
Voltage adds up
For
Two resistors in parallel in a circuit are arranged as follows:

For Resistors in Parallel:
Voltage is the same
Current adds up
For
There are many way in which resistors can be connected so that the circuits formed cannot be reduced to a single equivalent resistor. Therefore, we use Kirchhoff's Rules:
Juction Rule The sum of currents entering a juction must equal the sum of currents leaving the juction.
Loop Rule The sum of potential differences across all the elements around any closed loop equals zero.
Possible Loop Rule Cases:
Loop in the same direction as battery (negative to positive)

Loop in the opposite direction as battery (negative to positive)

Loop in the same direction as resistor (current is to the right)

Loop in the opposite direction as resistor (current is to the right)

RC circuits are ones which has a resistor and a capacitor.
Let's consider DC circuits containing capacitors. In these circuits, current varies with time. As the capacitor becomes more charged, the rate at which it is able to receive charge (current) decreases.
The maximum charge capacity of a capacitor is:
If we assume that the capacitor is uncharged before the circuit is on, the charge on the capacitor at time
Charge at time
A capacitor charges very slow with a long time constant.
A capacitor charges very fast with a short time constant.
If we assume that the capacitor is charged with a charge
Charge at time
Electric companies distribute electric power through a pair of wires. One of them is connected to the ground, and the other (called the hot wire) is at a potential of
Electrical devices in a home are connected in parallel.
A circuit breaker (or fuse) is connected in series with the wire entering the home. If too much is connected to the power source and the wires get too hot due to the current, the circuit breaker pops the fuse, disconnecting the entire circuit, and preventing a possible fire.
Some appliances require
For safety, many plugs have a third wire, called a case ground. This is so that in case another wire fails, there is still a grounding wire. If there wasn't, something else must act at the ground wire (a person).
Current can be measure with ammeters.
Voltage can be measure with voltmeters.
An ideal ammeter would have a very low resistance
An ideal voltmeters would have a very high resistance
There are a lot of different important equations regarding electricity. Here is a PDF document with all the electricity equations you need to memorize in one. Click the link above or the image below.
The ultimate source of magnetic fields is electric current.
The two ends of a magnet are called the poles.
Like poles repel, and unlike poles attract.
Poles of a magnet cannot be isolated like singular charges can.
Magnetism can be induced.
An unmagnetized piece of iron can be magnetized by stroking it with a magnet.
There are 2 types of magnetic materials:
Soft Magnetic Materials These are easily magnetized but also lose their magnetism easily.
Hard Magnetic Materials These are difficult to magnetize but also retain their magnetism better.
The direction of a magnetic field at any location is the direction in which the north pole of a compass would point.
When a charged particle is moving through a magnetic field, a magnetic force acts on it. The particle must be moving for there to be a force. The equation for magnetic force is the following (
The SI units of
The direction of the magnetic force is always perpendicular to both
The Right-Hand Rule:

The right-hand rule can be used to determine the directions of everything. Everything shown above has right angles between them.
Direction of Magnetic Fields:
Crosses are used to mark magnetic fields going into the page.
Dots are used to mark magnetic fields going out of the page.

Magnetic forces are also exerted on a current-carrying wire, similarly to a single charged particle.
Let's take a wire with length
This means that the total force would be represented by:
Remember that
If a current travels through a loop and is placed inside a magnetic field, a torque will be exerted on the loop as the result of a magnetic force.

In the diagram above, the forces on side

When calculated, the total torque is:
This only works if everything is perpendicular. Let
The torque is also dependent on the amount of times the wire is coiled. This number is denoted by
For short, the expression
Consider a positively charged particle moving through a magnetic field in a way such that the velocity of the particle is always perpendicular to the magnetic field. Since the force is perpendicular to both the magnetic field and the velocity, it acts as a centripetal force.
In 1819, Oersted found that en electric current in a wire caused there to be an electric field around the wire. The magnetic field goes in a circle around the wire. Using another right hand rule, it is possible to figure out the direction of the magnetic field.
Point your thumb in the direction of the current through the wire. The direction that the rest of your fingers curl is the direction of the magnetic field around the wire.
The direction of the magnetic field at a given point is tangent to this circle.
Force on a Particle When acting on a particle, the force is:
Force on another Wire The force that a wire exerts on another wire is:
For two wires with current, they both exert magnetic forces on each other mutually and symmetrically.
Current in the same direction: attracts
Current in the opposite direction: repels
Due to the spins of electrons within materials, some materials, like metals, have electrons that don't have a pair counteracting the spin. This motion causes there to be a small magnetic field. However, each particle in a material usually has magnetic fields oriented in completely random directions, meaning that the overall magnetic field is near zero.
When put under a magnetic field, though, all the orientations can align, magnetizing the material.
A ferromagnetic material is one which has enough lone electrons within its atoms to be magnetized.
A magnetic domain is a large group of atoms with spins that are aligned.
In 1819, Hans Oersted discovered that an electric current exerted a force on a magnetic compass. Electric current can produce a magnetic field and vice versa.
Using an experiment, Michael Faraday figured out that a current can be created using a changing magnetic field. To quantify how a magnetic field changes, we use concept called magnetic flux.
The magnetic flux
Magnetic flux can be visualized by thinking of, at a certain angle, how many magnetic field lines can pass through a coil of wire.
Electric current is induced when there is a change in magnetic flux. In turn, an EMF is produced, which is described by:
The negative sign simply indicated the polarity of the induced emf and, therefore, the direction of the induced current. Since it is hard to visualize the direction from the equation, when calculating, the negative can be omitted to obtain the magnitude of the emf, and Lenz's Law can be used to find the direction.
Lenz's Law When there is a change in magnetic flux, a direction can be associated with it: through the loop forward or backward. The direction of current is in the direction that opposes the change in flux. Use the right hand rule to find the direction of current from the direction of opposing flux.
When a conductor moves through a magnetic field, an EMF is produced. Think of a metal bar moving to the right through a magnetic field directed into the page. Using the right hand rule, the motion of negative charges within the bar is downward. This means that a current is induced upward.
If the circuit is completed, this concept is analogous to increasing or decreasing the area of the closed circuit loop.
Remember that
Electromagnetic waves are composed of fluctuating electric and magnetic waves. All light is electromagnetic waves.
A transformer is used to either upscale or downscale a voltage.
A transformer works by magnetizing a conductive loop with a certain amount of coils. On the other side of the loop, the change in flux induces a different voltage. To always make sure the flux is changing, transformers only work with AC current.

We know that the induced voltage is
At first, this may seem like free energy, but, in fact, the power remains the same. This means that as the voltage varies, current also varies:
During the early stages of study, electric and magnetic fields were thought to be unrelated. In 1865, James Clerk Maxwell provided a mathematical theory that showed a close relationship between all electric and magnetic phenomena:
Electric field lines originate on positive charges and terminate on negative charges (free charges exist).
Magnetic field lines always form closed loops (free charges do not exist).
A varying magnetic field induces an emf and hence an electric field.
Magnetic fields are generated by moving charges.
Electromagnetic waves are waves in which electric and magnetic fields traverse space in sync.
In an antenna, if AC current is running through it, the circuit loses energy in the form of electromagnetic waves. In fact, any circuit with AC current radiates EM waves.
As the charges move back and forth, an electric field parallel to the antenna is produced and radiates outwards. Additionally, a magnetic field perpendicular to the antenna is also produced.
Plane wave - a transverse wave that oscillates in one plane.
Electromagnetic waves travel at the speed of light.
Electromagnetic waves are transverse waves.
The ratio of the electric field to the magnetic field in an electromagnetic waves is the speed of light.
Electromagnetic waves carry both energy and momentum.
Because all electromagnetic waves travel through space at the speed of light, their wavelength and frequency are related by:

This is the spectrum of EM waves.
Important unit notes:
Equations and Concepts Review PDF
Light has a dual nature. In some cases, it acts like a wave, in others, it acts like a particle.
Newton explained light as a stream of particles. Huygens showed that a wave theory of light could also explain the laws of reflection and refraction. At the time, all waves needed a medium to travel through, and, if light were a wave, it would bend around corners (light does actually bend around objects (diffraction)).
Over the course of many experiments, it was concluded that light acts both as a wave and as a particle.
The energy of a photon of light is proportional to its frequency.
When light traveling in a transparent medium encounters a boundary, part of the incident ray is reflected.
Specular Reflection The reflection of light from a smooth surface
Diffuse Reflection The reflection of light from a rough surface

The law of reflection states that the reflected ray has the same angle to the normal with which the incident ray hits the boundary.
When light traveling through a transparent medium encounters another transparent medium, part of the ray is refracted and enters the second medium.

The angle of refraction,
High speed to low speed: bends toward normal
Low speed to high speed: bends away from normal
The index of refraction,
As light travels from one medium to another, its frequency must stay the same, and, using the formula
Combining this with the previous equation, we obtain Snell's Law of Refraction:
The index of refraction of a material actually depends on the wavelength of light entering it. This is called dispersion. The index of refraction decreases with increasing wavelength. This means that the longer the wavelength of light, the less it refracts.
A prism uses this to separate light into its respective colors that make it up.
In a prism, the angle between the entering and leaving waves is called the angle of deviation, denoted by

When light enters a medium in which it bends away from the normal, at some angle called the critical angle, the refracted beam will travel along the surface of the material. At every angle beyond this, the light will completely reflect back inside of the material. This is called total internal reflection. This angle can be found using Snell's Law:
Mirrors reflect
Lenses refract
On object placed in front of a mirror is at point
Images are formed at the point where rays of light intersect.
Real Image In a real image, light actually passes through the image point.
Virtual Image In a virtual image, light does not pass through the image point, but appears to come from there.
Images have three qualities:
Magnification The magnification is the ratio of the image height to the actually height:
If
Virtual or Real Difference discussed above.
Orientation: Upright or Inverted
If
The center of curvature of a spherical mirror is denoted by
The object
The object is inside point
When the light rays come in close to parallel, the reflected light rays can converge, however, if they are not close to parallel, they do not converge, forming blurry image known as a spherical aberration.
The magnification is:
The mirror equation, where
This can be rewritten as:
The focal length represents where the image of an object that is infinitely far away appears.

In ray diagrams, there need to be at least two (preferably three) rays drawn.
The first ray must come in parallel and pass through the focal point.
The second ray must pass through the focal point and exit parallel.
The third ray must reflect back on itself.
A real image is formed where the rays converge. If they seemingly don't converge, by extending the rays, a hypothetical convergence point can be found. This represents a virtual image being formed.
The Sun appears to be above the horizon even after it has actually set. This is because air does have a slight index of refraction, meaning it bends light toward the ground in the case of the Sun, slightly.
A mirage is formed by hot air. This can be seen as the "shimmer" above a long stretch of road on a hot day. As the road warms up, so does the air around it, and warm air has a different index of refraction. The light from an object traveling toward the ground can bend upward and can be seen as a "reflection".
Lenses are two-sided pieces of glass that bend light in certain ways. The two sides can consist of convex, concave, or straight edges.
Converging lenses are thickest in the middle.
Diverging lenses are thickest at the edges.
The equations for lenses are the same as those for mirrors.
A converging lens has a positive focal length.
A diverging lens has a negative focal length.
The focal length for a lens in air is related to the curvatures of its front and back surfaces and to the index of refraction

When multiple lenses are used one after the other, the image formed by the first lens is treated as the object for the second lens.
Lenses and mirrors aren't perfect, so the light may not converge exactly at a point, sometimes producing a blurry-looking image. This is called a spherical aberration.
Similarly, different color light has different wavelengths, meaning that they get refracted in slightly different ways. This is called a chromatic aberration. The image on the left was captured with a camera that does not correct chromatic aberrations, and the image on the right has the aberration corrected. Notice the color shifts near edges.

Geometric optics, covered by the last chapter, depends on the particle nature of light. Wave optics, however, depend on the wave nature of light.
2 waves can add together constructively or destructively. For interference between two sources of light to be observed, the following conditions must be met:
The sources must be coherent, which means the waves they emit must maintain a constant phase with respect to each other.
The waves must have identical wavelengths.
You can also hear interference. This is how noise-cancelling headphones work. They emit a sound which attempts to cancel out background noise by interfering with it destructively.
This is what the double slit experiment looked like:

The visible patters consists of bright and dark parallel bands called fringes.
Let's say the two slits are separated by a distance
For the following two equations,
Constructive Interference:
Destructive Interference:
The distance between the fringes can also be obtained.
If
Bright Fringes:
Dark Fringes:
When a light wave reflects from a surface the two possible cases are possible.
If the
If the
In summary, an electromagnetic wave undergoes a phase change of
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The diagram above shows how interference can happen in films. Depending on whether or not the film has a higher or lower index of refraction compared to the mediums either above or below it, the light in either point may reflect with or without a phase change.
If the thickness of the film is
The equation for constructive interference is:
The equation for destructive interference is:
When the thickness of a film varies, it can result in interesting patters, such as rings:
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Huygen's Principle - Every point on a wave form acts as a source of tiny wavelets that move forward with the same speed as the wave.
Huygen's Principle means that, when light passes through any opening similar to the size of the wavelength of the light itself, an interference pattern emerges.
This is the phenomenon mentioned above. Because every point of the light's path acts as a source of waves, light from one portion of the slit can interfere with light from another portion.
The equation for destructive interference for single-slit diffraction is:
Additionally, the points of constructive interference occur halfway between points of destructive interference.
The resulting interference patterns from single-slits and double-slits are very similar. However, there are some key differences.
Width of the 0th order bright fringe In a double-slit interference pattern, the width of the 0th order bright fringe is the same as the width of any other bright fringe. However, in a single-slit interference pattern, the width of the 0th order bright fringe is twice that of all the other bright fringes.
Rate of intensity with respect to distance The rate at which the intensity of light decreases as you move away from the center is greater for a single-slit interference pattern than it is for a double-slit interference pattern. In other words, the intensity of the bright fringes gets weaker faster for single-slit interference patterns than it does for double-slit interference patterns.
The diffraction grating is a device that analyzes a light source by using a large number of equally spaced parallel slits. The condition for maxima in the interference pattern, with
The interference pattern for a diffraction grating differs slightly compared to a double slit interference pattern, shown below:
Double Slit Interference Pattern Example:

Diffraction Grating Interference Pattern Equivalent:

Equations and Concepts Review PDF
Newtonian physics fails when considering particles moving close to the speed of light. The concepts of special relativity which explains this often violates our common sense about dimensions/time. It is, however, verified by particle accelerators all throughout the world.
The speed of any particle with mass must be less than the speed of light. Furthurmore, the concepts of force, momentum, and energy do not apply for rapidly moving objects. Observers moving at different speed will measure time and displacements differently.
In order to be able to describe a phsyical event, it's necessary to choose a frame of reference. The Principle of Galilean Relativity suggests that the laws of mechanics must be the same in all inertial frames of reference.
According to Einsein, the speed of light is the same for all observers.
If this is the case, though, then Galilean Relativity doesn't work. Imagine this: you're in a car that is moving at the speed of light and you turn your headlights on. What happens? If light were to appear, then it would be moving at twice the speed of light relative to an outside observer. In other words, there would be no constant speed of light.
The addition laws of velocity are therefore different.
In 1905, Einstein proposed a theory which solved the contradiction of the velocity addition. However, his proposal altered the way we think about space and time.
The two postulates of relativity:
All the laws of physics are the same in all inertial frames.
The speed of light in a vacuum is always
In relativistic mechanics, there is no such thing as absolute length or absolute time.
Common sense indicates that time passes just as quickly for someone on the ground than someone moving in a space craft. In contrast, special relativity reveals that the person on the ground measures time passing more slowly than the person in the space craft.
Suppose there are two light clocks, as shown in the picture below:

The light for the scenaria in which the object is moving seems to travel a longer distance in the same amount of time, but that cannot be. Therefore, time is dilated.
According to the astronaut in the spacecraft, the time interval
For the observer on the ground, however, the light travels a longer distnace. Therefore, they must measure a greater time interval. In other words, the Earth observer finds that the astronaut's clock runs slowly. If the speed of the spacecraft is
Combining the two equations, we get the formula for time dilation:
Because of time dilation, events that are simulaneous in one frame of reference may not be simultaneous in another.
Because time is different depeding on your frame of reference, since the speed remains constant, distance is therefore also different. Based off of
Length contraction only occurs in the direction of motion.
Apart from time and distances changing, momentum and energy also changes at exteremely high speeds.
Relativistic momentum is always larger than nonrelativistic momentum.
Think back to how the traditional addition of velocities worked. At low speeds, this works. However, at high speeds, the addition of velocities changes also. This is because everything must stay relative to and relatively below light speed.
One of Einstein's principles is the mass-energy equivalence principle. This means that mass and energy are interchangable. The rest energy of something with mass is:
For an object that is moving, its total energy is:
Using theses two values, their difference is the kinetic energy of an object at high speeds:
Using this formula, we can see that as
The relationship between total relativistic energy and momentum is:
When a particle is at rest,
For particles that have no mass, such as photons, the equation becomes:
In 1900, Planck developed a formula for blackbody radiation, a perfect emitter that has no energy losses. He proposed that blackbody radiation was produced by submicroscopic charged oscillators, called resonators.
The resonators were allowed to only have certain discrete (quantized) energy levels,
The photoelectric effect entails that light incident on certain metallic surfaces cause the emission of electrons from the surfaces. The emitted electrons are called photoelectrons.
Photoelectric Apparatus:
An evacuated glass tube called a photocell contains a metal plate (the emitter) connected to the negative terminal of an ammeter.
Another metal plate (the collector) is connected to the positive terminal of the ammeter.
When the tube is dark, the ammeter reads
A successful explanation of the photoelectric effect was given by Einstein:
A tiny packet of light energy (called a photon) would be emitted when an electron jumped down an energy level.
Conservation of energy means that the decrease in energy of the oscillator is equal to the energy of the photon.
When coming into contact with certain metals, if a photon carries enough energy, it could transfer its energy to an electron and eject it from the material.
The work required for this depends on how strongly the electron is held.
For the least strongly held electrons the necessary work has a minimum value of
If a photon has excess energy, it is transferred into the electron as kinetic energy. Therefore, the electrons that are held onto the weakest have the most kinetic energy.
Einstein proposed the following equation using conservation of energy:
Experimentally, a linear relationship is observed between
The cutoff wavelength is the minimum wavelength of light required to eject an electron. Referring to the previous equation,
Applications:
A very useful application of this effect are photocells (solar panels). Think about how your scientific calculator works.
Roentgen discovered x-rays in 1895. X-rays are a type of light, so they travel at the speed of light, and they are not charged particles.
In 1912, Max von Laue suggested that, if x-rays really were a form of light with really small wavelengths, it should be possible to diffract them by using the regular atomic spacings of a crystal lattice as a diffraction granting.
Production of X-Rays X-rays are produced when high speed electrons are suddenly slowed down (like when a metal target is struck by electrons that have been accelerated through several thousand volts).

Below is the graph of X-Ray Intensity per Unit Wavelength vs. Wavelength:

Notice the peaks. These are called characteristic x-rays because they are characteristic of the target material.
As electrons at high speeds approach a surface, they decelerate due to positively charged nuclei in the material. This is represented by the Bremsstrahlung curve on the graph. Sometimes, the electron has the perfect conditions to eject another electron from the
The electron that decelerated lost kinetic energy, which is now carried by the x-ray photon. In a very extreme and unrealistic case where all of an electron's kinetic energy is lost, the energy carried by the x-ray photon would be:
In theory, the wavelength of an electromagnetic wave can be measured if a grating having a suitable line spacing can be found. The spacing between the lines must be approximately equal to the wavelength.
The wavelength of x-rays happens to approximately equal the spacing between atoms. The structure of the atoms is determined by analyzing the positions and intensities of the various spots in the pattern.

In a crystal structure, an x-ray that's deflected on a lower layer of the material travels a farther distance than one that is deflected on a higher layer.

The reflected beams will combine and produce constructive interference when the path difference is an integer multiple of the wavelength. This can be seen by Bragg's Law:
In 1923, Arthur Compton found out that, when x-rays were directed at a block of graphite, the scatter waves had lower energy. The amount of energy lost depended on the angle at which the rays scattered. This is called the Compton shift.
Compton Shift - The change in wavelength
Compton thought that an x-ray photon carries both energy and momentum. As the x-ray photon comes into contact with a specific material, some of its energy and momentum is transferred into an electron in the material.
In chapter 26, we showed the following:
By manipulating these two equations, we can get the following equation for momentum:
In the case of lightwaves, however,
Also remember that
The shift in wavelength of a photon is given by:
The Photoelectric Effect and Compton Effect offer evidence that light is a particle. However, since light exhibits interference and diffraction, it also acts as a wave.
Because of this, light is said to have a dual nature, having both wave and particle characteristics at once.
In 1924, de Broglie thought that since light waves exhibit particle characteristic, perhaps all forms of matter have a dual nature. This lead to the fact that electrons have a dual nature, too.
De Broglie's Hypothesis:
The dual nature of matter is quite apparent, as the equation contains both particle concepts and wave concepts.
This experiment accidentally proved that electrons behave as waves.
The Heisenberg Uncertainty Principle - If the measurement of the position of a particle is made with precision
All this pretty much says is that it is not possible to exactly precisely measure the position and momentum of a particle simultaneously.
Another form of this equation sets a limit on the accuracy of the measurement of energy within a time interval
John Dalton's Model The billiard ball model - tiny, hard, indestructible spheres
J.J. Thomson's Model The plum pudding model - electrons embedded in a large positively charged cloud
Earnest Rutherford's Model The nuclear model - a nucleus with electrons orbiting around it
Rutherford's model has a problem. With the electrons undergoing centripetal acceleration, they radiate electromagnetic waves, losing energy in the process. Eventually, the electrons will fall into the nucleus and the atom would collapse.
Emission Spectra An evacuated glass tube filled with a gas with low pressure with will radiate light when a voltage is applied across it. When the light is analyzed with a spectrometer, an emission spectrum can be observed. Each element's emission spectrum is unique.
In 1885, when Balmer observed the emission spectrum of hydrogen, he found that the wavelengths of the lines can be described using the following equation, where
He didn't know it, but the 2 in
Apart from emitting light, an element can also absorb light at specific wavelengths, resulting in it having an absorption spectrum. The wavelengths of absorption spectra are the same as the wavelengths of emission spectra.
In 1913, Niels Bohr came up with a theory that explains why certain elements produce certain emission spectra. His theory applies to the hydrogen atom:
The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction.
Only certain orbits are stable.
Radiation is emitted when the electron jumps from a high-energy state to a ground state.
The size of allowed electron orbits is determined by a condition imposed on the electron's orbital angular momentum. The allowed orbits are where the angular momentum of the electron is an integer multiple of
This formula comes from the fact that Bohr said electrons travel in orbits. While in orbit, though, they have a wavelength. The position of the electron at the beginning of an orbit must match up with its position at the end of the orbit.

The circumference of a circle with radius
Remember the de Broglie wavelength:
The electric potential of the atom is represented by:
Simplifying, we get:
Assuming that the nucleus is at rest, the the total energy of the atoms is
Looking at just the electron, the attractive force is the centripetal force:
Plugging this value for kinetic energy back into the previous equation for energy, we get:
Combining more of the equations in very arbitrary ways that are not worthy of a derivation, we end up with the formula for the radius of orbit of an electron:
This is basically saying that electrons can only exist in certain allowed orbits determined by the integer
Plugging this
This is the energy corresponding to an energy level.
Ionization Energy Ionization energy is the energy required to completely remove an electron from the ground state.
Plugging this equation for energy into
Depending on which level electrons jump to and from, a different type of light is emitted. Each different type has a name.
Balmer Series - Visible Light
Lyman Series - UV Light
Paschen Series - Lower Energy Light
Quantum mechanics is in agreement with classical physics as long as energy differences between quantized levels are very small. Basically, physics is weird inside an atom.
Nuclei consist of protons and neutrons
The atomic number
The neutron number
The mass number
The notation for nuclei, where
For example, the notation for helium with 2 neutrons and 2 protons is
Isotopes Atoms with the same element but a different number of neutrons.
The atomic mass of an element is the weighted average of all the naturally occurring isotopes of an element.
Relative Mass
Because the rest energy of a particle is given by
A nucleus consisting of positively charge particles doesn't fall apart because of the nuclear force. This is an attractive, short-range force that acts between all nuclear particles. The protons attract each other through the nuclear force and repel each other through the Coulomb force.
The nuclear forces between proton-proton, proton-neutron, and neutron-neutron interactions are approximately the same.
As an atom gets more and more protons, it requires more neutrons to hold it together in order to overpower the Coulomb force. Because of this, as the elements get heavier on the periodic table, the more neutrons they have compared to protons.
The total mass of a nucleus is always less than the combined mass of the particles that make it up (nucleons). This difference in mass

In 1896, Becquerel accidentally discovered the radioactivity of uranium.
Radioactivity is the spontaneous emission of radiation from a nucleus.
Experiments that scientists of that time conducted suggested that radioactivity is the result of the decay (disintegration) of unstable nuclei. The rules of radioactivity are governed by the mass-energy equivalence: particles decay only when their combined products have less mass after decay than before (energy comes out).
Rutherford
Rutherford showed that there are 3 types of radiation: alpha (
Alpha Particles (
Alpha particles are relatively weak. They can barely penetrate a sheet of paper. Even when traveling through air, it can only travel a few centimeters, as it grabs onto stray electrons and becomes a harmless helium atom.
Beta Particles (
Beta-plus decay:
Beta particles are a little stronger. They can penetrate a few millimeters of aluminum and eventually lose their energy through collisions.
Gamma Particles (
The three types of radiation can be observed by putting the radioactive sample in a magnetic field:

This is a stability curve:

The green line represents the point at which
Alpha Particles By losing 2 neutrons and 2 protons, elements in the alpha particle area of the graph eventually hit the stability line.
Beta-minus Emitters By losing an electron and "gaining a proton," beta-minus emitters take the most direct path to reach the stability line.
Beta-plus Emitters By gaining an electron and "losing a proton," beta-plus emitters take the most direct path to reach the stability line.
The rates of radioactive decay appear to be absolutely constant, even under extreme circumstances. Observation has shown that if a radioactive sample contains
Introducing the decay constant
Alternatively, this can be viewed as a exponential differential equation:
Isotopes with a large
A unit of activity is a curie (
The SI unit of activity is a becquerel (
The time it takes for a sample to decay to half of its original mass is called its half life.
Solving the above mentioned differential equation with the initial condition
Turning this into an exponential decay function by a factor of
If a nucleus emits an alpha particle, it loses 2 protons and 2 neutrons. The original nucleus is called the parent nucleus, and the nucleus after decay is called the daughter nucleus.
When one element changes into another, it is called spontaneous decay or transmutation.
The parent must contain more mass than the alpha particle and daughter combined as the excess mass is transformed into kinetic energy.
When a nucleus undergoes beta decay, the daughter nucleus has the same number of nucleons, but the atomic number is changes by 1.
The Neutrino
When an atom undergoes radioactive decay, the kinetic energy due to change in mass is actually less than it should be. This energy goes to a particle that we don't really see, called a neutrino. The neutrino (
The nucleus, like electrons, only exists in quantized energy states. When a nucleus undergoes radioactive decay, it is left in an excited state. When the nucleus transitions from this high energy state to a lower energy state, a high energy photon (gamma ray) is emitted.
Gamma decay does not cause transmutation, and there's really no way to predict it.
The top problems section is an assortment of the most challenging notes problems, homework problems, lab problems, FRQs, and some external problems I found on the internet to help review for each chapter.
If you are able to solve all the problems without issue, you are probably substantially prepared for the test.
I tried to include a variety of problems that hit all topics covered in each chapter.
Some questions may by slightly edited to fit the format of this document.
CJ Eq. of Kinem. for Const. Acc. Q6 A speed ramp at an airport is basically a large conveyor belt on which you can stand and be moved along. The belt of one ramp moves at a constant speed such that a person who stands still on it leaves the ramp
CJ Eq. of Kinem. for Const. Acc. Q9 A locomotive is accelerating at
CH 2 WS #4 Q11 A peregrine falcon dives at a pigeon. The falcon starts downward from rest and falls with free-all acceleration. If the pigeon is
CJ Proj. Mot. Q7 The drawing shows two planes each about to drop an empty fuel tank. At the moment of release each plane has the same speed of 
CJ Proj. Mot. Q11 A rocket is fired at a speed of
CH 3 WS #3 Q8 Two canoeists in identical canoes exert the same effort paddling and hence maintain the same speed relative to the water. One paddles directly upstream (and moves upstream), whereas the other paddles directly downstream. If downstream is the positive direction, an observer on shore determines the velocities of the two canoes to be
CH 2&3 Review Sup. Prob. Q19 A river has a steady speed of
CH 4 WS #1 Q14 A
CH 4 WS #3 Q7 A 
CH 4 WS #5 Q8 Two blocks of masses
CH 4 Test Review Q16 A toboggan slides down a hill and has a constant velocity. The angle of the hill is
CJ Cons. of E. Q14 Two pole-vaulters just clear the bar at the same height. The first lands at a speed of
CH 5 WS #4 Q9 A surprising demonstration involves dropping an egg from a third-floor window so that the egg lands on a foam-rubber pad without breaking. If a
CH 5 WS #5 Q7 A
CH 5 WS #6 Q7 While running, a person dissipates about
CH 5 WS #6 Q11 A
CH 5 WS #1 Q10 A
CH 5 WS #2 Q14 A
CH 6 WS #2 Q13 A
CH 6 WS #4 Q5 A
CH 6 Supplemental Problems Q4 An estimated force-time curve for a baseball struck by a bat is shown in the figure. From this curve, determine
(a) the impulse delivered to the ball and (
CJ Chosen Tan. V&A Q15 A person lowers a bucket into a well by turning the hand crank, as the drawing illustrates. The crank handle moves with a constant tangential speed of 
CH 7 WS #2 Q4 An air puck of mass 
CJ Chosen Cent. F, Banked Curves, G Q15 A stone has a mass of 
CJ Chosen Cent. F, Banked Curves, G Q17 On a banked race track, the smallest circular path on which cars can move has a radius of 
CJ Chosen Cent. F, Banked Curves, G Q21 Saturn has an equatorial radius of
CH 7 WS #3 Q15 A pail of water is rotated in a vertical circle of radius
CH 7 Notes P14 Q5 A projectile is fired straight up from the South Pole of Earth with an initial speed
CH 7 Notes P15 Q6 How much energy is required to lift a
CH 7 Notes P15 Q7 Compute the escape velocity (in
CH 7 Notes P17 Q1 A satellite of mass
External If Saturn is, on average,
External An object starts from rest at point 
CH 8 WS #1 Q11 The person in the figure weighs 
CJ Chosen Rot. Equil. Q11 The drawing shows a person (weight, 
CH 8 WS #3 Q8 A water molecule consists of an oxygen atom with two hydrogen atoms bound to it as shown in the figure. The bonds are 
CH 8 WS #4 Q9 A 
CH 8 WS #5 Q11 In the figure, the sliding block has a mass of 
CH 8 WS #6 Q10 A playground merry-go-round of radius
CH 8 WS #6 Q15 A
External A traffic light hangs from a pole as shown in the figure. The uniform aluminum pole 
External If the mass of Mars is
AP Mechanics Review 2 Q9c A box of uniform density weighing
2022 Practice Exam 3 MCQ Q9 Two objects,
a.
CH 9 Notes P13 Q1 A solid, square, pinewood raft measures
CH 9 Notes P17 Q2 In the condition known as atherosclerosis, a deposit or atheroma forms on the arterial wall and reduces the opening through which blood can flow. In the carotid artery in the neck, blood flows three times faster through a partially blocked region than it does through an unobstructed region. Determine the ratio of the effective radii of the artery at the two places. (
CH 9 Notes P20 Q1 A nearsighted sheriff fires at a cattle rustler with his trusty six-shooter. Fortunately for the cattle rustler, the bullet misses him and penetrates the town water tank to cause a leak. If the top of the tank is open to the atmosphere, determine the speed at which the water leaves the hole when the water level is
CH 9 Notes P22 Q1 An aneurysm is an abnormal enlargement of a blood vessel such as the aorta. Suppose that, because of an aneurysm, the cross sectional area
CH 9 WS #3 Q8 An empty rubber balloon has a mass of
CH 9 WS #4 Q9 A fountain sends a stream of water
CH 9 WS #4 Q12 A hypodermic syringe contains a medicine with the density of water (see figure). The barrel of the syringe has a cross-sectional area of 
External When a golf ball with a radius of
CH 10 Notes P9 Q3 A sealed glass bottle at
CH 10 Notes P9 Q5 Verify that one mole of oxygen occupies a volume of
CH 10 Notes P14 Q1 Air is primarily a mixture of nitrogen,
CH 10 Notes P14 Q2 In the previous example, we found that the average translational kinetic energy of each molecule in air is
CH 10 Notes P8 Q1 In scuba diving, a greater water pressure acts on a diver at greater depths. The air pressure inside the body cavities (e.g. lungs, sinuses) must be maintained at the same pressure as that of the surrounding water, otherwise they might collapse. A special valve automatically adjusts the pressure of the air breathed from a scuba combustion tank to ensure that the air pressure equals the water pressure at all times. There is some scuba gear that consists of a
CH 10 WS #3 Q7 The total random translational kinetic energy of the water molecules in a
External A pressure of
CH 11 WS #1 Q10 A
CH 11 WS #2 Q2 A large block of ice at
CH 11 WS #2 Q4 What mass of steam that is initially at
CH 12 Notes P3 Q1 A gas is contained in a cylinder with a moveable piston on top. The gas is at a pressure of
CH 12 Notes P4 Q4 Gas in a cylinder moves a piston with area
CH 12 Notes P5 Q1 A system absorbs
CH 12 Notes P6 Q3 The temperature of three moles of a monatomic ideal gas is reduced from
CH 12 Notes P11 Q2 Find the efficiency of an engine that introduces
CH 12 Notes P14 Q1 Water near the surface of a tropical ocean has a temperature of
CH 12 Notes P15 Q4 The highest theoretical efficiency of a gasoline engine, based on the Carnot cycle, is
CJ Chosen 1st Law Prob. Q8 Three moles of an ideal monatomic gas are at a temperature of
CH 12 Notes P19 Q1
CH 12 WS #3 Q5 Two
CH 13 Notes P8 Q1 A
CH 13 Notes P15 Q1 AM and FM radio waves are transverse waves consisting of electric and magnetic disturbances traveling at a speed of
CH 13 Notes P17 Q1 A uniform string has a mass 
CH 13 Notes P17 Q2 To what tension must a string with mass
CH 13 WS #3 Q14 A light string mass 
CH 15 Notes P3 Q1 The electron and proton of a hydrogen atom are separated (on average) by a distance of about
CH 15 Notes P4 Q3 Three charges lie along the x-axis as in the figure. The positive charge
CH 15 WS #1 Q12 A typical lead-acid storage battery contains sulfuric acid,
CH 15 Notes P6 Q8 In the Bohr model of the hydrogen atom, the electron is in a circular orbit about the nuclear proton at a radius of
CH 15 Notes P8 Q1 Tiny droplets of oil acquire a small negative charge while dropping through a vacuum (pressure
CH 15 WS #4 Q11 Each of the protons in a particle beam has a kinetic energy of
CH 16 Notes P4 Q6 In atom smashers (also known as cyclotrons and linear accelerators), charged particles are accelerated in much the same way they are accelerated in TV tubes: through potential differences. Suppose a proton (mass
CH 16 Notes P5 Q7 Suppose electrons (mass
CH 16 Notes P10 Q5 Three point charges are initially infinitely far apart. They are then brought together and placed at the corners of an equilateral triangle. Each side of the triangle has a length of 
CH 16 Notes P16 Q4
(a) Calculate the equivalent capacitance between 
CH 16 Notes P17 Q1 A fully charged capacitor defibrillator contains
CH 16 WS #4 Q6 A parallel-plate capacitor has
CH 17 Notes P1 Q2 Suppose
CH 17 WS #1 Q11 A teapot with a surface area of
CH 17 Notes P9 Q3 An electric heater is operated by applying a potential difference of
CH 18 WS #2 Q7 Two
CH 18 WS #2 Q8 An unmarked battery has an unknown internal resistance. If the battery is connected to a fresh
CH 18 WS #2 Q12 Determine the potential difference, 
CH 18 Notes P12 Q1 An uncharged capacitor and a resistor are connected in series to a battery. If
AP Electricity Review 2 Q6e The circuit above contains a battery with negligible resistance, a closed switch
CH 19 WS #2 Q13 Consider the mass spectrometer shown schematically in the figure. The electric field between the plates of the velocity selector is 
CH 19 Notes P15 Q2 Two wires, each having a weight per unit length of
CH 19 WS #3 Q6 The figure is a cross-sectional view of a coaxial cable (such as a VCR cable). The center conductor surrounded by a rubber layer, which is surrounded by an outer conductor, which is surrounded by another layer of rubber. The current in the inner conductor is 
CH 20 WS #1 Q9 A circular wire loop of radius
CH 20 WS #1 Q15 A square, single-turn coil
CH 20 Notes P7 Q1 An airplane with a wingspan of
CH 20 Notes P8 Q3 The sliding bar in the following figure has a length of 
CH 20 WS #2 Q3 A circular coil, enclosing an area of 
2017 International Practice Exam MCQ Q9 A rectangular loop of copper wire is attached to a cart by an insulating rod. The cart is moving at constant speed when it enters a region containing a uniform magnetic field that is perpendicular to the plane of the loop and directed into the page, as shown below. Frictional losses are negligible. Which of the following correctly describes the speed of the cart as it moves into, through, and out of the field? (d)
a. The speed remains constant.
b. The speed continually decreases.
c. The speed decreases as the cart enters the field and increases as it leaves the field but is constant while it is completely inside the field.
d. The speed decreases as the cart enters and leaves the field but is constant while it is completely inside the field.
CH 20 WS #3/CH 21 WS #1 Q10 A transformer on a pole near a factory steps the voltage down from
CH 22 Notes P7 Q5 Light of wavelength
CH 22 WS #1 Q14 The laws of refraction and reflection are the same for sound as for light. The speed of sound is
CH 22 WS #2 Q3 A narrow beam of ultrasonic waves reflects off the liver tumor in the figure. If the speed of the wave is 
CH 22 WS #2 Q6 A submarine is
CH 23 Notes P 5 Q3 Assume that a certain concave spherical mirror has a focal length of
CH 23 WS #1 Q4 A dentist uses a mirror to examine a tooth. The tooth is
CH 23 WS #1 Q6 A dedicated sports car enthusiast polishes the inside and outside surfaces of a hubcap that is a section of a sphere. When he looks into one side of the hubcap, he sees an image of his face
CH 23 WS #2 Q7 A man standing
CH 23 WS #4 Q11 A person looks at a gem with a converging lens with a focal length of
CH 23 WS #4 Q13 An object is placed
CH 24 Notes P8 Q5&6 A pair of glass slides
CH 24 WS #3 Q7 A screen is placed
CH 26 WS #1 Q6 The average lifetime of a pi meson in its own frame of reference is
CH 26 WS #2 Q4 A muon formed high in the Earth’s atmosphere travels at speed
CH 26 WS #2 Q5 A spaceship of proper length
CH 27 Notes In converting electrical energy into light energy, a
CH 27 Notes Light of wavelength
CH 27 Notes An electron (
CH 27 Summary WS Q3 The threshold of dark-adapted (scotopic) vision is
CH 27 Summary WS Q5 Ultraviolet light is incident normally on the surface of a certain substance. The binding energy of the electrons in this substance is
CH 27 Summary WS Q11 The spacing between planes of nickel atoms in a nickel crystal is
CH 27 Summary WS Q18 After learning about de Broglie’s hypothesis that particles of momentum
CH 27 Summary WS Q20 A
CH 28 Summary WS Q4 The “size” of the atom in Rutherford’s model is about
CH 28 Summary WS Q7
(a) If an electron makes a transition form the
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